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 BEBR 
 
 FACULTY WORKING 
 PAPER NO. 1356 
 
 Estimation Risk and Incentive Contracts for 
 Portfolio Managers 
 
 Susan I. Cohen 
 Laura T. Starks 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois, Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1356 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 May 1987 
 
 Estimation Risk and Incentive Contracts for Portfolio Managers 
 
 Susan I. Cohen, Associate Professor 
 Department of Business Administration 
 
 Laura T. Starks 
 University of Texas 
 
Abstract 
 
 The fiduciary relationship between portfolio managers and the 
 investors they represent may be viewed as a principal-agent relation- 
 ship, and therefore we have used the methodology from the agency 
 literature in economics and finance to study the impact of existing 
 compensation arrangements on the conflicts of interest between these 
 two groups. In this paper we employ the assumptions of the Capital 
 Asset Pricing Model and of estimation risk concerning beta to develop 
 a model in which portfolio managers can, through effort, choose the 
 parameters of the beta distribution. Our model entails moral hazard 
 because the investor cannot observe the manager's action. Given 
 certain utility functions we show that the presence of estimation risk. 
 leads the manager to choose a lower beta portfolio than otherwise and 
 that no first best optimal contract exists. We also show that the 
 direction of the managerial divergent behavior is related to the 
 divergence in risk preferences between the manager and the investor. 
 By making slightly more stringent assumptions about preferences, we 
 show that there exist some conditions under which the manager will 
 provide more effort but also a riskier portfolio than the investor 
 prefers. Finally we show that the investor will prefer a manager who 
 is less risk averse than the investor. 
 
I. Introduction 
 
 The conflict of interest between investment advisors and their 
 clients concerns economists, policy makers, and investors, among others. 
 Of particular interest is the way in which compensation arrangements 
 between the two parties affect the conflict. The fiduciary relation- 
 ship between portfolio managers and the investors they represent may 
 
 be viewed as a principal-agent relationship, and therefore we have used 
 
 2 
 the methodology from the agency literature in economics and finance to 
 
 study the impact of existing compensation arrangements on the conflicts 
 
 of interest between these two groups. 
 
 In the relationship between portfolio managers and shareholders, as 
 in all agency relationships, problems arise due to the absence of com- 
 plete information. The investor cannot costlessly observe all the 
 actions of the portfolio manager. In particular, the investor cannot 
 observe the returns distribution of the chosen portfolio. Since the 
 manager is motivated to choose a portfolio which is optimal in terms of 
 his or her own welfare, the investor wants to know whether such a choice 
 is optimal in terms of the investor's welfare. The problem is further 
 complicated if there is uncertainty associated with the parameters of 
 the distribution of the portfolio returns. If the manager can lower 
 this "estimation risk" through the provision of effort, there are two 
 potential sources of divergent behavior on the part of the manager: 
 the amount of effort and the portfolio characteristics. 
 
 The above problems could possibly be resolved if the investor were 
 able to observe portfolio outcomes over time, and as a consequence, 
 
-2- 
 
 infer the underlying stochastic process generating these outcomes. It 
 might then be possible for the investor to eliminate the costs of past 
 divergent behavior by requiring the manager to "settle up" ex post . 
 However, ex post settling up is not costless, since it would most likely 
 involve some kind of legal action. In addition, complete settling up 
 is not possible if the manager is bankrupt. Consequently, here, as in 
 all principal-agent relationships, the investor will prefer to avoid 
 such problems by setting up incentive compensation contracts. These 
 contracts are used to motivate the portfolio manager to take actions in 
 the investor's best interest. 
 
 We develop a model which describes the portfolio choices made by 
 the manager. We make several assumptions about both the returns distri- 
 bution and what the manager can do. The driving assumption of this 
 model is that the manager can choose the distribution of the parameters 
 of the returns distribution of the portfolio, and that the parameters' 
 distributions have non-zero variances. For given means of the parame- 
 ters' distributions, the manager, can through the provision of effort, 
 reduce the variance of these distributions. 
 
 We assume that expected returns follow the Capital Asset Pricing 
 Model. In the absence of "estimation risk," the portfolio manager would 
 choose the beta for the portfolio. When there is estimation risk, the 
 manager chooses the mean and the variance of the distribution of beta. 
 Expected utility for both the manager and the investor are then func- 
 tions of the mean and the variance of beta; we assume that these 
 
 functions are, additively separable in the mean and the variance, concave 
 
 3 
 
 in the mean, and decreasing in the variance. Under these conditions, 
 
-3- 
 
 we show a moral hazard problem will always exist. However, unlike many 
 other principal-agent models in which it is not possible to eliminate 
 the moral hazard problem, our model includes conditions under which the 
 agent actually provides more effort than the investor desires in 
 equilibrium, but chooses a variance which is not optimal from the 
 investor's perspective. Under these conditions we still have divergent 
 behavior, but of a different type. 
 
 In contrast, Starks [1987] showed that without uncertainty about 
 the parameters of the returns distribution of the portfolio, and with 
 no effort provided by the manager, an optimal linear sharing rule may 
 exist between the manager and the investor. However, if the manager is 
 able to use effort to increase the expected return of the portfolio, 
 that is, change the mean of the returns distribution without changing 
 the variance, a linear sharing rule will not be optimal. In that case 
 the portfolio manager will choose a level of effort that is less than 
 the investor wishes in equilibrium, but will choose the variance that 
 the investor wishes. That result depends crucially on the assumption 
 that the manager can choose the mean and the variance of the returns 
 distribution independently. In our model, the results depend crucially 
 on the assumption that the manager's choices of the mean and of the 
 variance of the portfolio are interdependent. 
 
 In the next section we formally present our assumptions and the 
 model. Section III contains the statement of our results (proofs are in 
 the Appendix). Conclusions are contained in Section IV. 
 
-4- 
 
 II. The Model 
 
 Consider a representative investor who turns over a sum of money, 
 K, to a representative portfolio manager for some period of time. After 
 the investor and manager agree on how the manager is to be compensated, 
 the manager selects a portfolio which earns some random return, R, over 
 
 the period. At the end of the period the portfolio is liquidated, the 
 
 4 5 
 manager is paid, and the residual goes to the investor. ' 
 
 Also assume that asset returns are normally distributed and that 
 
 they are modelled according to the Sharpe-Lintner-Mossin Capital Asset 
 
 Pricing Model (CAPM), which implies that expected returns can be written 
 
 6 
 as: 
 
 (1) E(R.) = R + BJE(R ) - RJ 
 i I l m r 
 
 where 
 
 R ■ the gross return on the investment portfolio i, 
 
 R = the gross return on the market portfolio, 
 m 
 
 R = the gross return on the riskfree asset, and 
 
 8. = the systematic relationship between R and R . 
 i i m 
 
 Since R is distributed normally, 
 
 (2) R i = E(R ) + a ± z 
 
 where z ~ N(0,1). 
 Substituting from (1), equation (2) becomes 
 
-5- 
 
 (3) R. = R f + b.[E(R ) - R. + or (z/r.) 
 
 i t 1 m r m l 
 
 where 
 
 r. = the correlation coefficient between the returns of portfolio 
 i 
 
 i and the market portfolio. 
 
 If the investor invests the amount K in the portfolio, then the ending 
 dollar value of the portfolio is 
 
 (4) x. = R.K. 
 
 We further assume that the investor and the manager agree that the 
 portfolio is to be well diversified and that the appropriate benchmark. 
 
 Q 
 
 of performance is the market portfolio. 
 
 The driving assumption of our model is that the true value of 8. 
 cannot be measured with certainty by the manager. However, we also 
 assume that the manager can reduce this "estimation risk" by providing 
 more effort. For example, the manager may search for more information 
 about the securities in the portfolio and thereby reduce the estimation 
 risk. (See Barry and Brown [1984, 1985] for further discussion.) 
 Formally, 
 
 8 is a random variable with mean b and variance 
 
 »i 2 
 
 where a is a measure of the manager's effort 
 
 used in estimating the true 6. • 
 Thus, we assume that as the mean risk level increases, the manager must 
 expend more effort in order to reduce the variance of the distribution 
 
-6- 
 
 9 
 of (3. • In addition, we assume that the market risk, of the portfolio, 
 
 z, is independent of the estimation risk.; that is, z and 8. are 
 
 independent random variables. 
 
 Since the investor cannot observe the true beta of the chosen port- 
 folio, nor the mean and variance of the distribution of beta, the 
 manager's compensation is a function only of what is jointly observable: 
 the ending value of the portfolio. Let S(x) denote the compensation 
 schedule. 
 
 Our main purpose in this paper is to describe the incentives created 
 by many existing compensation contracts, not to (necessarily) define an 
 "optimal" contract. Most existing contracts are linear functions of 
 the portfolio returns. For example, the most common contract in current 
 use pays the manager a percentage of the portfolio value. Another 
 popular contract is the symmetric performance contract, which can be 
 written as a linear function of both the portfolio and the market re- 
 turns. Consequently we restrict our attention to linear contracts. 
 
 The manager's utility is assumed to be a function of the effort 
 expended in reducing beta estimation risk as well as the manager's 
 
 wealth W , which itself is a function of the compensation schedule, 
 M 
 
 S(x ). Specifically, the manager's utility function is additively 
 separable in these variables, concave in the wealth variable and convex 
 decreasing in effort. Similarly, the investor is assumed to have a 
 utility function which is concave in wealth. 
 
 Given linear fee contracts, from equations (4) and (1) it can be 
 seen that both the manager's and investor's wealth can be written as 
 
-7- 
 
 linear functions of beta. Thus, for expositional purposes we can talk 
 about the individual's expected utility with respect to beta. 
 
 We also restrict our attention to certain types of utility of 
 
 wealth functions, specifically those whose expectations are concave in 
 
 2 
 the mean and decreasing in the variance of beta. Defining u and a as 
 
 the mean and variance of beta, respectively, we employ the following 
 
 , ,_ * , 12,13 
 expected utility of wealth function: 
 
 (5) E fl [U (8)J = F (u) - G (a 2 ) 
 
 p • • • 
 
 for all p, a ; where 
 
 (i) there exists a value of u such that F = 
 (li) F < 
 
 « 
 
 (iii) g' > 
 (iv) g" > 
 (v) G (0) = 0. 
 
 Equation (5) incorporates both a utility function and a fee contract. 
 It is the result of having taken expectations with respect to both 
 sources of uncertainty: market risk and estimation risk. 
 
 With our previous assumption of linear contracts, there are several 
 conceivable utility functions in wealth which would provide the above 
 expectation. One example is the quadratic utility function. However, 
 our results do not depend on a quadratic utility of wealth function. 
 In the usual CAPM world, (that is when there is no estimation risk and 
 the variance of 8 is zero), if returns are normally distributed then 
 expected utility is concave in 8 with an interior maximum. For our 
 
-8- 
 
 2 
 model, when a = 0, expected utility is assumed to equal F ( y) , a 
 
 concave function of y with an interior maximum. Since the distribu- 
 tion of returns is assumed to be normal, and the distribution of 6 is 
 
 now degenerate at y, we have the usual CAPM world. The additional 
 
 2 
 structure that we require is that when a * 0, expected utility can be 
 
 written as additively separable in the mean and variance of 8. 
 
 Since managers maximize expected utility, they will not agree to a 
 
 compensation schedule unless they can be assured the expected utility 
 
 to be gained from their next best opportunity. Let Y denote the 
 
 representative manager's minimum required expected utility. If the 
 
 contract (compensation schedule) is accepted, the manager will choose 
 
 a mean and a variance (and thus a portfolio) such that the manager's 
 
 expected utility is maximized. 
 
 2 
 In the absence of uncertainty about beta, a - and utility depends 
 
 only upon the mean of the degenerate distribution of beta. In this case 
 
 the manager would choose a value of b such that: 
 
 (6) F M (b) = 0. 
 
 Denote this value by b . Similarly, in the absence of uncertainty about 
 beta and ignoring the reservation wage of the manager, the investor's 
 most preferred beta would solve: 
 
 (7) F*(b) = 0. 
 
-9- 
 
 Denote this value by b T . 
 
 We define the comparative risk preferences of the manager and the 
 investor as follows. 
 
 Definition 
 
 The manager will be said to be less, equally, or more risk, averse 
 
 2 
 as compared to the investor at (u, a ) as 
 
 F M F i 
 
 — - is greater than, equal to, or less than — p, respectively. 
 
 G G 
 
 M I 
 
 This definition just states that the more risk averse individual requires 
 a greater increase in the mean of the beta distribution for an equivalent 
 increase in the variance. 
 
 Definition 
 
 The manager will be said to be globally less, globally equally, or 
 
 globally more risk averse as compared to the investor if 
 
 F 1 F' 
 
 M I 2 
 
 — r is greater than, equal to, or less than — - for all (u, a ). 
 
 °M G I 
 
 The results of the paper depend on the relationship between the 
 manager's and the investor's risk preferences. In the next section we 
 show that the relationship between the investor's most preferred values 
 of b and a (and thus mean and variance) and the manager's choices 
 depends upon the relative risk preferences of the manager and the 
 investor. 
 
-10- 
 
 III. Results 
 
 The relationship between the risk preferences of the manager and of 
 
 the Investor has an impact on the choices that these Individuals would 
 
 2 
 make in the absence of uncertainty. In particular, when a = 0, the 
 
 less risk averse individual chooses a higher value for b, i.e., chooses 
 
 a less risky portfolio. It is straightforward to demonstrate that if 
 
 the manager is globally less, equally or more risk averse than the 
 
 investor, then b.. is greater than, equal to or less than b T respectively. 
 M 1 
 
 This relationship is consistent with the usual definition of risk 
 
 aversion in a CAPM world. 
 
 We now move on to an examination of the manager's problem. We 
 
 assume that the manager's expected utility is separable In wealth and 
 
 effort where g(a) is the manager's disutility of effort; g' > and 
 
 g" > 0> Given the assumptions of our model the manager will choose a 
 
 return-risk-effort combination by choosing b and a so as to solve: 
 
 h 2 
 (7) MAX F M (b) - G M (|-) - g(a). 
 b,a 
 
 The first order conditions for the manager's problem are then: 
 
 (8) f>)-<(f-)-^-0. and 
 
 2 2 
 
 Ma l 
 a 
 
 Since the manager's utility function is concave in b and a, conditions 
 (8) and (9) are both necessary and sufficient. Let (b*,a*) denote the 
 solution to the manager's problem. We have implicitly assumed that the 
 
-11- 
 
 raanager's expected utility is strictly greater than Y, the manager's 
 
 reservation wage. If the manager's expected utility at (b*,a*) was 
 
 just equal to Y, any moral hazard problems would be trivially eliminated. 
 
 Our first Proposition states that without estimation risk the 
 manager would prefer a higher beta portfolio; that is, the mean of the 
 distribution of beta chosen by the manager will be strictly less than 
 the manager's most preferred beta in the absence of estimation risk. 
 
 Proposition 1: b* < b when a* > 0, and variance is positive. 
 
 C JJ 
 
 The intuition behind this result is straightforward. Estimation 
 risk provides another source of risk for the manager's chosen portfolio. 
 Since the manager's risk tolerance takes both estimation risk and 
 systematic risk into consideration, if one is eliminated, the manager is 
 willing to assume more of the other. Thus, if estimation risk does not 
 exist, then the manager will take on more systematic risk. Note that 
 this proposition is not exclusive to the manager, it also holds for the 
 investor. Thus, ceteris paribus , both parties will want a risker port- 
 folio if there is no estimation risk. 
 
 We next consider the potential moral hazard problem between the 
 investor and the manager. It can be eliminated if a first-best contract 
 for the investor exists. This will be the case if the solution (b*,a*) 
 to the manager's problem is also a solution to: 
 
 h 2 
 (10) MAX F x (b) - G I (|-) 
 
 b,a 
 
 h 2 
 
 ••*• F M (b > - Vr " g(a) -^ Y * 
 
-12- 
 
 We call this the investor's problem and denote its solution by (b,a) 
 The first order conditions for the investor's problem are: 
 
 2 2 
 
 ' ' b 2b ' ' b 2b 
 
 (11) Ffb) - G T (— ). -^ + X[F M (b) - Gf-5-). H] = 0, 
 
 I laa M Maa 
 
 2 2 2 2 
 
 (12) g[(— ). \+ ^ G m ( — >• \" *'<*>] = 0. 
 
 a a 
 
 h 2 
 
 (13) X[F M (b) - G M (^-) - g(a) - Y] = 0. 
 
 Again, by concavity, these conditions are both necessary and suf- 
 ficient. Since the objective function is strictly increasing in a, we 
 know that the constraint will be binding and therefore that X > 0. We 
 can then rewrite (13) as: 
 
 h 2 
 
 (14) F M (b) - G(— ) - g(a) - Y = 0. 
 M Ma 
 
 Our second proposition states that given our assumptions there is 
 no contract that eliminates the moral hazard problem inherent in the 
 manager's portfolio selection process. Thus, there is no way to 
 compensate the manager to ensure that he or she makes a decision that 
 is optimal from the investor's perspective. 
 
 Proposition 2 : No first-best optimal contract exists. 
 
 We demonstrate this Proposition (in the Appendix) by showing that 
 
 for all F , G , F and G satisfying our initial assumptions, (b*,a*) * 
 I L M M 
 
 (b,a). 
 
-13- 
 
 Turning our attention to the impact of existing compensation 
 contracts on the choices made by the manager, we note that a primary 
 concern is the severity of the moral hazard problem inherent in these 
 contracts. One factor which affects the problem is the relationship 
 between the risk, tolerance of the manager and that of the investor. We 
 now consider what types of investor-manager combinations mitigate the 
 problem. 
 
 If the manager is at least as risk averse as the investor, then a 
 common moral hazard problem occurs in which the manager provides less 
 effort and chooses a less risky portfolio than the investor prefers. 
 The intuition is clear: the manager chooses less effort (smaller a) 
 because the manager must pay for it and the investor need not. This 
 leads the manager to trade off to a lower mean of the beta distribution 
 as well, since that is the only other way to reduce variance. 
 
 On the other hand, if the manager is less risk averse than the 
 investor, then conditions may exist under which the manager is willing 
 to expend more effort than the investor prefers. In fact, we show that 
 a necessary condition for the manager to provide more than the investor's 
 preferred effort is that the manager is globally less risk averse than 
 the investor. 
 
 Theorem 
 
 (1) If the manager is globally more or globally equally risk averse 
 than the investor, then b* < b and a* < a. 
 
 (2) If a _< a* , then b <^ b* and the manager is globally less risk averse 
 than the investor. 
 
-14- 
 
 The most interesting result can be obtained as a Corollary to the 
 Theorem when we place more structure on the manager and investor 
 utility functions. 
 
 We assume that: 
 
 2 2 2 
 
 (15) F ] .(u) - G ] .(a ) = a n u + a^u + a^a + c^ 
 
 where a < and a > 0. 
 C16) F M (w) " G n (a 2 ) = o^^ 2 + <^ 2 u + c^^ + c^ 
 
 where a,. < and ol,~ > 0. 
 
 "Ml ^ U and ^2 
 
 The F and G functions will have such a form if the expected utility 
 with respect to market uncertainty (systematic risk) is a quadratic 
 function of beta. 
 
 When the expected utility has the above representation, the dif- 
 ference in risk preferences between the manager and investor can be 
 characterized completely by b and b . That is, the manager is 
 globablly less risk averse than the investor if and only if b > b ; 
 globally equally as risk averse if and only if b = b . Under this 
 additional structure, we have the following Corollary: 
 
-15- 
 
 Corollary 
 
 (1) If b M < b T , then b* < b and a* < a. 
 
 (2) For each b M , there exist b T and b T , where b T < b T < b.., such that 
 
 (a) For be(b\,b w ), b* < b and a* < a; 
 I I M 
 
 (b) For b I e(b I ,b ] .], b* '> b and a* < a; and 
 
 (c) For b e(-»,b ], b* > b and a* > a. 
 
 In no case does the manager make the choice combination that the 
 investor would prefer. The manager either chooses less effort and less 
 risk than the investor would like, less effort and more risk, or more 
 effort and more risk. The last result is a natural consequence of the 
 assumption that as the beta of a portfolio increases, the estimation 
 risk increases. Thus, the manager must expend more effort to get the 
 same variance around the chosen mean as he or she would have to for a 
 smaller beta. Since the investor desires a smaller beta, he or she 
 will also be satisfied with less effort put into reducing the estimation 
 risk surrounding that beta. Note that in this last case the managerial 
 divergent behavior is not in the choosing of too much effort, but in 
 the choosing of too much risk. Although the investor would be satisfied 
 with less effort, he or she is not averse to the manager putting in 
 more effort than would be required. 
 
 What implications do this Theorem and Corollary have for the 
 investor in choosing a portfolio manager? Since the contracts in 
 current use do not seem to protect the investor against managerial 
 divergent behavior (considering the compensation schedule only), the 
 
-16- 
 
 investor may consider alternative strategies to mitigate this problem. 
 For example, the investor could choose a manager with greater risk, 
 tolerance and then invest a portion of his or her funds into a risk free 
 (unmanaged) portfolio in order to hedge the higher risk the manager has 
 presumably undertaken. In this situation, the investor needs to know 
 the risk preferences of the various portfolio managers in order to 
 choose one with higher risk tolerance. With mutual fund managers such 
 knowledge is possible due to the advertising of the portfolio's invest- 
 ment objective. This objective has been found to be a good indicator 
 of the risk of mutual fund portfolios. 
 
 Given our assumptions of linear contracts and quadratic expected 
 utility functions, we can make more definitive statements concerning 
 the investor's preference over managers. For expository purposes, take 
 one of the simplest forms of linear contracts (and the most commonly 
 used form), s«x, where the manager receives a percentage of the final 
 market value of the portfolio assets. For any fraction s, suppose the 
 investor could search over different managers and choose one with the 
 
 desired b w . Thus, the investor could choose the manager whose b w is 
 M M 
 
 "best" from the investor's perspective; let this "best" b w be denoted 
 
 M 
 
 by b . Since this action is equivalent to choosing b (with b fixed) 
 so as to maximize the investor's expected utility (under the constraint 
 that the manager maximizes his or her own expected utility), we can 
 make a statement concerning the investor's choice of manager. 
 
-17- 
 Proposition 3 : The investor will choose a manager such that b > b . 
 
 If the investor had the ability to choose any manager for a given 
 contract s, the investor would choose a manager who is globally less 
 risk, averse than the investor. Since this holds for every contract s, 
 it must also hold for the second-best optimal contract. That is, since 
 the choice of the second-best contract is just the choice of s, and 
 thus implicitly b M and b. , we can make a statement about such a contract, 
 
 Proposition 4 : The investor prefers a manager and compensation contract 
 combination in which the manager is globally less risk averse than the 
 investor. 
 
 The intutition behind this result is that if the manager is enough 
 less risk averse than the investor, the manager may be willing to supply 
 
 extra effort in order to make a "more risky" portfolio acceptable to the 
 
 1 f\ 
 investor. This result is consistent with Ross' [1979] partial 
 
 equilibrium analysis of the market for agents in which he shows that 
 less risk averse agents can outbid more risk averse agents in contract- 
 ing with any risk averse principal. 
 
 One aspect of this final result that should be noted is that the 
 difference between the investor's and the manager's preferences over 
 beta depends in part upon the manager's aversion to effort, which is 
 captured in the g(») function. If the manager is sufficiently less 
 risk averse than the investor, the manager will provide more than the 
 investor's first best level of effort, and choose a portfolio with more 
 systematic risk than the investor's first best choice. 
 
-13- 
 
 IV. Conclusions 
 
 In this paper we have shown the impact of estimation risk on the 
 portfolio manager-investor problem when the investor does not have 
 complete information concerning the manager's actions. In our model 
 under the assumptions of the (1) Capital Asset Pricing Model, and (2) 
 estimation risk, concerning beta, portfolio managers are able to choose 
 the parameters of the beta distribution. Given this model the managers 
 are shown to choose portfolios with less systematic risk than they would 
 if no estimation risk existed. We also show that no linear contract in 
 our world can be first best. That is, the manager will not choose the 
 same amount of effort or risk as the investor would prefer. Since the 
 investor cannot observe the manager's actions, a moral hazard problem 
 exists. 
 
 Previous literature in agency theory has generally proved that when 
 a moral hazard problem exists, the agent provides less effort than the 
 principal would prefer. However, our results show that the direction 
 of the moral hazard problem depends on the relationship between the 
 risk tolerance of the manager and that of the investor. That is, there 
 exist some situations in which the manager may provide more effort than 
 the investor would prefer. However, the manager also chooses a riskier 
 portfolio than the investor would prefer. We show that given the most 
 common compensation contract in use today, the investor prefers a 
 manager whose risk choice is higher than the investor's own. Thus, the 
 relationship between these risk tolerances must be an important factor 
 in the selection of a portfolio manager. By taking this relationship 
 into consideration, the investor may be able to mitigate the inherent 
 moral hazard problem. 
 
-19- 
 
 Although our results are developed under specific assumptions 
 regarding the manager's and investor's utility functions, these 
 assumptions are not as restrictive as they may at first appear. The 
 important assumption is that expected utility is concave in beta. 
 There are a number of reasonable utility functions that are consistent 
 with this assumption. Thus, we believe our results are quite 
 generalizable. 
 
-20- 
 
 Footnotes 
 
 For example, see Wall Street Journal , November 15, 1985, p. 7, 
 "SEC to Let Investment Advisers Collect Performance Fees From Certain 
 Clients" and Wall Street Journal , January 9, 1986, p. 1, "Business 
 Bulletin." 
 
 2 
 For example, agency theory has been used to study the insurance 
 
 relationship (see Zeckhauser [1970], Spence and Zeckhauser [1971], and 
 
 Shavell [1979]); the employer-employee relationship (see Stiglitz 
 
 [1974, 1975] and Harris and Raviv [1978]); and the management -owner 
 
 relationship (see Jensen and Meckling [1976]). 
 
 3 
 For example, if utility is quadratic in wealth or returns are 
 
 normally distributed, and contracts are linear in returns, then 
 expected utility will have the stated properties. 
 
 4 
 Given these conditions we are assuming that the relationship 
 
 between the investor and the manager is static rather than dynamic. The 
 
 major rationale for this assumption is the tractability it supplies. 
 
 In our model we ignore the ability of the investor or the manager 
 
 to reduce risk through diversification. The investor is assumed to 
 
 employ only one manager and the manager is assumed to work for only one 
 investor. 
 
 Although investors believe that assets are priced according to 
 CAPM, it is not assumed that CAPM holds absolutely. There must be some 
 type of market imperfection in order for the posited fiduciary relation- 
 ship to exist. Otherwise, there would be no need for the portfolio 
 manager, and the individual investor could select the portfolio him or 
 herself. Note that our results would not change with a single factor 
 arbitrage pricing model. 
 
 This assumption is not terribly restrictive since Farrar (1962) 
 and others have found that most mutual funds are well diversified. 
 Formally, this assumption implies that r = 1 for all portfolios. 
 
 o 
 
 Problems due to the fact that the true market portfolio may not be 
 observable are ignored here. 
 
 9 
 
 This assumption is consistent with the traditional means of 
 
 estimating beta through time series analysis. In other words, the 
 estimation of the portfolio beta becomes more difficult as the average 
 beta increases. For example, a portfolio with beta close to zero could 
 be achieved with most of the investment in riskfree assets and a few 
 risky assets. Only the risky assets need to have beta estimated. As 
 the desired beta increases, the portfolio must include more risky 
 assets. Thus, the greater the number of assets whose betas must be 
 estimated, the more variance there will be in the portfolio beta. 
 
-21- 
 
 In addition, as the desired beta increases, the opportunity set 
 decreases. Consequently, there is less information on which to base 
 estimates. Again, the variance of the portfolio beta will be higher. 
 Therefore, as the desired portfolio beta increases, the manager must 
 provide more and more effort to maintain the same variance of the 
 estimate. 
 
 See Starks [1987] for a more detailed discussion of the kinds of 
 contracts that are used between investors and mutual fund managers, as 
 well as a discussion of the legal restrictions placed on the kinds of 
 contracts that can be used. 
 
 In addition, some portfolio managers (including mutual fund 
 managers) are restricted by law to linear compensation contracts. 
 
 12 
 
 As our concern here is not with how a particular contract is 
 
 arrived at, bargaining and negotiation between the investor and the 
 
 manager is ignored. 
 
 13 
 
 In the rest of the paper we delete the subscript i referring to 
 
 the portfolio. A choice of mean and variance is a choice of portfolio. 
 
 14 
 
 We employ quadratic expected utility because of its mathematical 
 
 simplicity. Since we are assuming normal distributions for market 
 returns, and since we are interested in the local properties of the 
 preferences of the manager and the investor, a quadratic expected 
 utility function provides us with a good local approximation of many 
 other concave utility functions with interior maximums. 
 
 In fact, the Investment Company Act of 1940 requires that mutual 
 fund managers make this disclosure. 
 
 Notice that since the investor knows the manager's preferences, 
 the investor then has the ability to reduce his or her own risk exposure 
 by investing more funds in riskfree assets. 
 
 However, in Ross' model there is no aversion to effort by the 
 
 agent. 
 
-22- 
 
 Appendix 
 
 Proof of Proposition 1 
 
 Since (1, > for positive variance, we have (from the first order 
 
 conditions (9) and (10)) that F (b*) > 0. Using F < 0, we get b* < b . 
 
 MM M 
 
 Proof of Proposition 2 
 
 A contract would be first best if there existed a (b,a) pair that 
 satisfied (8) - (14). Substituting (9) into (12) we have: 
 2 2 
 
 G ;<*-) . \ = o. 
 
 Ia I 
 a 
 
 2 2 2 
 
 Since < a < » (from (9)), either G^(|-) = or ^ = 0. If ^- = 0, 
 
 a a 
 
 then b = 0; the mean and the variance of beta are both zero. There is 
 
 no longer any estimation risk; nor is there any systematic risk. In 
 
 that case investors do not need managers, since they could just put all 
 
 b 2 
 of their investment in the risk-free asset. If b * 0, then — > 0. 
 
 2 a 
 
 • b 
 That gives us G (— ) > 0. Therefore, no (b,a) pair exists satisfying 
 I a 
 
 (8) - (14). 
 
 Proof of Theorem 
 
 We first demonstrate some preliminary results. Let (b.(a),a) solve: 
 
 , b*(a) 2b, (a) 
 
 (8«) F M (b.(a))- G M (-i ) * = 
 
 Ml Ma a 
 
 and (b_(a), a) solve: 
 
 2 2 
 
 , b ? (a) b 9 (a) 
 
 (9') G(— )• -~ g'(a) = 0. 
 
 Ma z 
 
-23- 
 
 Clearly, b* = b^a*) = b 2 (a*). 
 
 It is straightforward to verify that 
 
 db, db 9 db,(a*) db ? (a*) 
 
 -r- - > 0, -r^- > 0, -^ -. < 0, 
 
 da da da da 
 
 lim b (a) = lim b (a) = 
 a+0 a+0 
 
 lim b^a) = b M , 
 a ■*■<*> 
 
 and verification is left to the reader. 
 This give us : 
 
 b^a) > b (a) iff a < a* and 
 
 b^a) < b 2 (a) iff a > a*. 
 
 The investor's first order conditions (where "hats" indicate functions 
 evaluated at (b,a)) are: 
 
 r.«t\ ,£» ^' 2b ,*i * i 2b, 
 (11') F - G . — + X[F - G . — ] = 
 1 i- a M M a 
 
 (12-) G-.J+x^.J-g-l-O 
 
 (14,) F M- G M _g - Y '°- 
 
-24- 
 
 Combining (IT) and (12') gives: 
 
 8 l I la a 2 IM IM 
 
 The terra on the right-hand-side in brackets is greater than zero, equal 
 to zero, or less than zero as the manager is globally more, globally 
 equally, or globally less risk, averse as compared to the investor, 
 respectively. 
 
 Since G > and X > 0, we have that 
 
 . K 2 
 
 V j* - «' < °- 
 
 This gives us the result that 
 
 b < b 2 (a) 
 
 By assumption of the Theorem: 
 
 F G - G F > 0. 
 I M I M - 
 
 Using g' > 0, we get: 
 
 
 Combining this with (IT) and A > 0, yields 
 
 p' . g' 2b < 0. 
 
 M M a _ 
 
 Therefore, 
 
 b > b (a) 
 
-25- 
 
 Corabinlng b < b 9 (a) and b > b (a), and using the established properties 
 of b, ( •) and b„(»), we have our desired result: 
 
 a > a*. 
 
 Thi 
 
 s also gives b > b*, since both b,( # ) and b ( •) are increasing in a. 
 
 (2) If a.< a*, and b ? is increasing in a, we get b < b~(a) _< b 9 (a*), 
 or b < b*. The rest is just the contraposition of part I. 
 
 Proof of the Corollary 
 
 (1) Immediate from the Theorem. 
 
 (2) Using the newly defined F and G functions, we can rewrite the 
 investor's first order conditions as: 
 
 (11") -2b(l +t) + 2b T + y[-2b(l +-7) + 2bJ = 
 
 -1 -2 
 
 (12") ^+ y[^- "g"* (a)] = 
 
 3, 3. 
 
 -2 
 (14") -(b 2 + ^-) + 2b M b +\ 3 ~ 1(a) -7=0 
 
 _ cr — °-\J[2> — Y 
 
 where g = — a — •, a = and Y = 
 
 ^3 
 
 °M1 MJ ""Ml _a Ml 
 
 Combining (U") and (12") we get 
 
 b 2 (b x - b M ) = g*(a). a 2 [-b(l + -7) + bj 
 
-26- 
 
 We wish to fix t> M (and thus b* and a*) and see what happens to b and a 
 as we change the investor's risk, parameter b T . Totally differentiating 
 (11"), (12") and (14"), and using Cramers' rule to solve, we get 
 
 . -2(-?2 - g*(a))(-2b(l + 4) + 2b x ) 
 
 da a a 
 
 db x = [H| 
 
 b — * 2 
 
 . 2(\-g'(a)) Z 
 
 db a 
 
 dbj [Hi 
 
 where H is the Hessian matrix of second partial derivatives. Since 
 |h| > from concavity, we have 
 
 $L-> o. 
 db i 
 
 da 1 
 
 However, the sign of — — depends on the sign of -2b(l + — ) + 2b_ , which 
 
 db a I 
 
 in turn depends on the difference b, - b . Using our combined first 
 order conditions and conditions (11") and (12"), we can easily show 
 that if 
 
 (a) b T > b M , then ~- < 0. 
 
 1 M db 
 
 (b) b T = b M , then 4r- " 0. 
 
 i M db 
 
 (c) b_ < b M , then 4r- > 0. 
 
 i M db 
 
 When b = 0, the investor wants only risk free assets and does not need 
 the manager. Therefore, 
 
 lira b(b ) - 0, lira a(b ) = 0. 
 b +0 l b +0 l 
 
-27- 
 
 We thus know that a(») is a continuous function of b_ that reaches a 
 
 a maximum at b = b . We have alread demonstrated that when b < b , 
 I M I M 
 
 a(b ) > a* > 0. Therefore, there exists b < b such that 
 
 a(b x ) = a* 
 by the intermediate value theorem, and 
 
 a(b ) < a* for all b < b . 
 If a < a*, then b < b*, which gives us (2)(c) of the Corollary. 
 
 Again using the continuity of b(«), we know that there exists b , where 
 
 b T < b T < b u , such that 
 1 I M. 
 
 b(b) - b*. 
 
 Additionally, bCbj) < b*, aCbj.) > a* for all b such that b < "b < b 
 < h^. This gives us (2)(b) of the Corollary. Using the continuity of 
 b( •) and a( •) again, we know that 
 
 bCbj.) > b* and a(b r ) > a* 
 
 for all b T such that "b T < b T < b . This is (2) (a) of the Corollary. 
 I I I M 
 
 Proof of Propostlon 4 
 
 Recall that the manager chooses b and a by solving the first order 
 conditions: 
 
 -2b( 1 + -) + 2b M = 
 a M 
 
 b 2 
 a 
 
 - g*(a) = 0. 
 
-28- 
 
 The investor chooses b (a manager) by solving the constrained 
 maximization problem: 
 
 MAX - b 2 (l + -) + 2b -b + a TO 
 , a I 13 
 b M ,a,b 
 
 s.t. -2b(l + — ) + 2b =0 
 a M 
 
 h 2 
 a 
 
 Using the first contstraint to solve for b in terms of b and a, and 
 
 M 
 
 substituting into the second constraint and the objective function, we 
 can rewrite the investor's problem as: 
 
 2 2 
 may b M a (1+a) . ,. V , - 
 ^'T^a?'" +2b I(^) + ^ 
 
 b M 2 
 s.t. — -r - g'(a) = 0. 
 
 (l+a) Z 
 
 The Lagrangian for this problem is then: 
 L(b M ,a,X) = 
 
 FIT tV 2b i " V + x[ "^2- g ' Ca)1 
 
 The first order conditions for the investor's problem are then: 
 
 M 2 b 
 
 3b M (l+a) I M (1+a) 2 
 
 4~ - 1 ~2 t b M (2b T " b M )] + X[ " — ^ ~ g" (a)1 = ° 
 
 33 (l+a) 2 M l M (l+a) 3 
 
 b 2 
 
 3L M ,, . _ 
 
 = - g'(a) = 
 
 3X (l+a) 2 
 
-29- 
 
 Let (b,,a) denote the solution to the rewritten problem. If X < 0, then 
 M — 
 
 b - b > and 2b T - b < 0, or b. > b > 2b , which is a contradiction. 
 I M — I M — I — M — I 
 
 Therefore, X > and 
 
 b M >b I 
 our desired result. 
 
 Proof of Proposition 5 
 
 Immediate from Proposition 4 and the definition of a second-best 
 optimal contract. 
 
-30- 
 
 Bibliography 
 
 Barry, C. and S. Brown, "Differential Information and the Small Firm 
 
 Effect," Journal of Financial Economics , Volume 13 (June 1984), pp. 
 283-294. 
 
 Barry, C. and S. Brown, "Differential Information and Security Market 
 Equilibrium," Journal of Financial and Quantitative Analysis , 
 Volume 20 (December 1985), pp. 407-422. 
 
 Farrar, D. , The Investment Decision Under Uncertainty (Englewood Cliffs, 
 NJ, Prentice-Hall, 1962). 
 
 Harris, M. and A. Raviv, "Some Results on Incentive Contracts with 
 
 Applications to Education and Employment, Health Insurance, and Law 
 Enforcement," American Economic Review (March 1978), pp. 20-30. 
 
 Jensen, M. , and W. Meckling, "Theory of the Firm: Managerial Behavior, 
 Agency Costs, and Ownership Structure, "Journal of Financial 
 Economics , Volume 3 (October 1976), pp. 305-360. 
 
 Shavell, S. , "Risk Sharing and Incentives in the Principal and Agent 
 Relationship," Bell Journal of Economics , Volume 10 (Spring 1979), 
 pp. 55-73. 
 
 Spence, M. , and R. Zeckhauser, "Insurance, Information, and Individual 
 Action," American Economics Review , Volume 61 (May 1971), pp. 380- 
 387. 
 
 Starks , L. , "Performance Incentive Fees: An Agency Theoretic Approach," 
 Journal of Financial and Quantitative Analysis , forthcoming, March 
 1987. 
 
 Stiglitz, J., "Risk Sharing and Incentives in Sharecropping," Review of 
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 Zeckhauser, R. , "Medical Insurance: A Case Study of the Tradeoff 
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